#### Second-order lower radial tangent derivatives and applications to set-valued optimization

Xu et al. Journal of Inequalities and Applications
Second-order lower radial tangent derivatives and applications to set-valued optimization
Bihang Xu 2
Zhenhua Peng 0 1
Yihong Xu 0
0 Department of Mathematics, Nanchang University , Nanchang, 330031 , China
1 School of Mathematics and Statistics, Wuhan University , Wuhan, 430072 , China
2 School of Information Engineering, Nanchang University , Nanchang, 330031 , China
We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.
Henig efficiency; radial tangent derivative; set-valued optimization; optimality condition
1 Introduction
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gent derivatives of the objective function and the constraint function are not separated,
and thus the properties of the derivatives of the objective function are not easily obtained
from those of the constraint function.
On the other hand, some efficient points exhibit certain abnormal properties. To
eliminate such anomalous efficient points, various concepts of proper efficiency have been
introduced [–]. Henig [] introduced the concept of Henig efficiency, which is very
important for the study of set-valued optimization [, , , ].
In this paper, we introduce a new class of lower radial tangent cones and two new kinds
of second-order tangent sets, using which we introduce four new kinds of second-order
tangent derivatives. We discuss the properties of these second-order tangent derivatives,
using which we establish second-order necessary optimality conditions for a point pair to
be a Henig efficient element of a set-valued optimization problem.
2 Basic concepts
Throughout the paper, let X, Y , and Z be three real normed linear spaces, X , Y , and Z
denote the original points of X, Y , and Z, respectively. Let M be a nonempty subset of Y .
As usual, we denote the interior, closure, and cone hull of M by int M, cl M, and cone M,
respectively. The cone hull of M is defined by
dom F := x ∈ X : F(x) = ∅ ,
graph F := (x, y) ∈ X × Y : y ∈ F(x) ,
epi F := (x, y) ∈ X × Y : y ∈ F(x) + C .
Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A. The radial
tangent cone of A at xˆ, denoted by R(A, xˆ), is given by
Remark . Equation (.) is equivalent to
where N denotes the set of positive integers.
Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A. The
contingent cone of A at xˆ, denoted by T (A, xˆ), is given by
Remark . (See []) Equation (.) is equivalent to
T (A, xˆ, w) := v ∈ X : ∃tn → + and vn → v such that xˆ + tnw + tnvn ∈ A .
Definition . (See [, ]) Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and
(uˆ , vˆ) ∈ X × Y . The second-order composed contingent derivative of F at (xˆ, yˆ) in the
direction (uˆ , vˆ) is the set-valued map D F(xˆ, yˆ, uˆ , vˆ) : X → Y defined by
Definition . (See []) Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈
X × Y . The second-order contingent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the
set-valued map DF(xˆ, yˆ, uˆ , vˆ) : X → Y defined by
In the following, we introduce a new class of lower radial tangent cones and two new
kinds of second-order tangent sets.
Definition . Let Q be a nonempty subset of X × Y , and let (xˆ, yˆ) ∈ cl Q. The lower radial
tangent cone of Q at (xˆ, yˆ) is defined by
such that (xˆ + tnun, yˆ + tnvn) ∈ Q .
Definition . Let Q be a nonempty subset of X × Y , and let (xˆ, yˆ) ∈ cl Q. The
secondorder lower radial tangent set of Q at (xˆ, yˆ) in the direction (uˆ , vˆ), denoted by Rl(Q, (xˆ, yˆ),
(uˆ , vˆ)), is given by
Definition . Let A be a nonempty subset of X, and let xˆ ∈ cl A. The second-order radial
tangent set of A at xˆ in the direction w, denoted by R(A, xˆ, w), is given by
R(A, xˆ, w) := v ∈ X : ∃tn > and vn → v such that xˆ + tnw + tnvn ∈ A .
Example . Let R be the set of real numbers, X = Y = R, Q = {(– n , n ) : n = , , . . .} ∪
{(x, y) : x ≥ , y ≥ } ∪ {(–, –)}, and (xˆ, yˆ) = (uˆ , vˆ) = (, ). A direct calculation gives
Rl(Q, (, ), (, )) = {(x, y) : x > , y ≥ }, T (Q, (, ), (, )) = {(x, y) : x ≥ , y ≥ } ∪
{(x, ) : x < }, and R(Q, (, ), (, )) = {(x, y) : x ≥ , y ≥ } ∪ {(x, ) : x < } ∪ {(x, x) : x <
} ∪ n∞={λ(– n , n ) : λ > }.
3 The second-order lower radial tangent derivative
In this section, by virtue of the radial tangent cone, the second-order radial tangent set,
the lower radial tangent cone, and the second-order lower radial tangent set, we introduce
the concepts of the second-order radial composed tangent derivative, the second-order
radial tangent derivative, the second-order lower radial composed tangent derivative, and
the second-order lower radial tangent derivative for a set-valued map. Furthermore, we
discuss some important properties of the second-order lower radial composed tangent
derivative and the second-order lower radial tangent derivative.
Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F , and (uˆ , vˆ) ∈ X × Y .
The second-order radial composed tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ)
is the set-valued map R F(xˆ, yˆ, uˆ , vˆ) : X → Y defined by
If R(R(epi F, (xˆ, yˆ)), (uˆ , vˆ)) = ∅, then F is said to be second-order radial composed
derivable at (xˆ, yˆ) in the direction (uˆ , vˆ) or that the second-order radial composed tangent
derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) exists.
Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F , and (uˆ , vˆ) ∈ X × Y .
The second-order radial tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the
setvalued map RF(xˆ, yˆ, uˆ , vˆ) : X → Y defined by
If R(epi F, (xˆ, yˆ), (uˆ , vˆ)) = ∅, then F is called second-order radial derivable at (xˆ, yˆ) in the
direction (uˆ , vˆ) or that the second-order radial tangent derivative of F at (xˆ, yˆ) in the
direction (uˆ , vˆ) exists.
Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈ X × Y .
The second-order lower radial composed tangent derivative of F at (xˆ, yˆ) in the direction
(uˆ , vˆ) is the set-valued map Rl F(xˆ, yˆ, uˆ , vˆ) : X → Y defined by
If Rl(Rl(epi F, (xˆ, yˆ)), (uˆ , vˆ)) = ∅, then F is said to be second-order lower radial composed
derivable at (xˆ, yˆ) in the direction (uˆ , vˆ) or that the second-order lower radial composed
tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) exists.
Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈ X × Y .
The second-order lower radial tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the
set-valued map RlF(xˆ, yˆ, uˆ , vˆ) : X → Y defined by
If Rl(epi F, (xˆ, yˆ), (uˆ , vˆ)) = ∅, then F is called second-order lower radial derivable at (xˆ, yˆ)
in the direction (uˆ , vˆ) or that the second-order lower radial tangent derivative of F at (xˆ, yˆ)
in the direction (uˆ , vˆ) exists.
Proposition . Suppose that E ⊂ X and the second-order lower radial composed tangent
derivative of F : X → Y at (xˆ, yˆ) ∈ graph F in the direction (uˆ , vˆ) exists. Then
Rl F(xˆ, yˆ, uˆ , vˆ) R R(E, xˆ), uˆ
⊂ clcone clcone F(E) + C – yˆ – vˆ .
v ∈ Rl F(xˆ, yˆ, uˆ , vˆ)(u).
uˆ + tnun ∈ R(E, xˆ).
xˆ + tnkukn ∈ E.
Therefore, there exist sequences tnk > and ukn → uˆ + tnun such that
(uˆ + tnun, vˆ + tnvn) ∈ Rl epi F, (xˆ, yˆ) .
xˆ + tnkukn, yˆ + tnkvkn ∈ epi F,
Then, for the same tnk and ukn, there exists a sequence vkn → vˆ + tnvn such that
and, consequently,
v + t v as k
→ ˆ n n
→ ∞
, we obtain
n n ∈
n ∈
and, consequently,
→ ∞
, we get
n ∈
cone clcone
clcone clcone
and, consequently,
R F(x, y, u, v) R (E, x, u)
ˆ ˆ ˆ ˆ ˆ ˆ
l
clcone cone
F(E) + C – y – v .
ˆ ˆ
Proposition . Suppose that E
X and the second-order lower radial tangent derivative
of F : X
F in the direction (u, v) exists. Then
ˆ ˆ
Proof Let v
R F(x, y, u, v)(R (E, x, u)). Then there exists u
ˆ ˆ ˆ ˆ ˆ ˆ
l
R F(x, y, u, v) R R(E, x), u
ˆ ˆ ˆ ˆ ˆ ˆ
l
clcone clcone
F(E) + C – y – v .
ˆ ˆ
R F(x, y, u, v) = R
ˆ ˆ ˆ ˆ
l l
u such that
For such t and u , it follows from (.) that there exists a sequence v
n n
v such that
and, consequently,
n ∈
n ∈
n n ∈
n n ∈
n ∈
cone cone
→ ∞
, we get
clcone cone
R F(x, y, u, v) R (E, x, u)
ˆ ˆ ˆ ˆ
l
ˆ ˆ ⊂
clcone cone
F(E) + C – y – v .
ˆ ˆ
D F(x, y, u, v) R R(E, x), u
ˆ ˆ ˆ ˆ
ˆ ˆ ⊂
clcone clcone
R F(x, y, u, v) R R(E, x), u
ˆ ˆ ˆ ˆ
ˆ ˆ ⊂
clcone clcone
Remark . If we substitute D F(x, y, u, v) or R F(x, y, u, v) for R F(x, y, u, v) in
Proposiˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
l
tion ., then none of the inclusions
is necessarily true. If we substitute DF(xˆ, yˆ, uˆ , vˆ) or RF(xˆ, yˆ, uˆ , vˆ) for RlF(xˆ, yˆ, uˆ, vˆ) in
Proposition ., then none of the inclusions
is necessarily true, as is shown in the following example.
Example . Let R be the set of real numbers, X = Y = R, C = {t : t ≥ }, and E = {x : x ≥
}. Define the set-valued map F : X → Y by
F(x) =
{y : y ≥ } if x ≥ ,
√
{y : y ≥ x} otherwise.
R(E, ) = R R(E, ), = [, +∞),
D F(, , , –)(x) =
R F(, , , –)(x) =
Rl epi F, (, ) = (x, y) : x ∈ R, y ≥ ,
Rl Rl epi F, (, ) , (, –) = ∅,
Rl F(, , , –)(x) = ∅, x ∈ R.
D F(, , , –) R R(E, ),
= R F(, , , –) R R(E, ),
= R,
Rl F(, , , –) R R(E, ),
= ∅,
Then, the inclusion of Proposition . is true. However,
D F(xˆ, yˆ, uˆ , vˆ) R R(E, xˆ), uˆ
⊂ clcone clcone F(E) + C – yˆ – vˆ
D F(, , , )(x) = R F(, , , )(x) = R R
F, (, ), (, )
≥ }
≥ }
R F(, , , )(x) = R F(, , , )(x) = [, + ), x
∞
l l
F, (, ) = (x, y) : x
= R
= T
F, (, ), (, )
F, (, ) = R
= (x, y) : x > , y
(x, y) : x
R F(x, y, u, v) R R(E, x), u
ˆ ˆ ˆ ˆ ˆ ˆ
clcone clcone
F(E) + C – y – v .
ˆ ˆ
(ii) Let (x, y) = (, ), (u, v) = (, ). A direct calculation gives
ˆ ˆ ˆ ˆ
R R(E, ), = R (E, , ) = R(E, ) = [, + ),
∞
F, (, ) , (, ) = R R
F, (, ) , (, ) = T
F, (, ), (, )
D F(, , , ) R R(E, ), = R F(, , , ) R R(E, ),
= D F(, , , ) R (E, , )
= R F(, , , ) R (E, , ) = R,
R F(, , , ) R R(E, ),
l
= R F(, , , ) R (E, , ) = [, + ),
∞
l
clcone clcone
F(E) + C – y – v =
ˆ ˆ
clcone cone
F(E) + C – y – v = [, + ).
ˆ ˆ ∞
Then, the inclusions of Propositions . and . are true. However,
D F(x, y, u, v) R R(E, x), u
ˆ ˆ ˆ ˆ ˆ ˆ
R F(x, y, u, v) R R(E, x), u
ˆ ˆ ˆ ˆ ˆ ˆ
D F(x, y, u, v) R (E, x, u)
ˆ ˆ ˆ ˆ ˆ ˆ
clcone clcone
clcone clcone
clcone cone
F(E) + C – y – v ,
ˆ ˆ
F(E) + C – y – v ,
ˆ ˆ
F(E) + C – y – v ,
ˆ ˆ
R F(x, y, u, v) R (E, x, u)
ˆ ˆ ˆ ˆ ˆ ˆ
clcone cone
F(E) + C – y – v .
ˆ ˆ
4 Second-order necessary optimality conditions
Let F : X
Z
, and (F, G) : X
be defined by (F, G)(x) = F(x)
Consider the following optimization problem with set-valued maps:
min F(x),
s.t. G(x) ∩ (–D) = ∅, x ∈ X.
Definition . (See [, , ]) Let xˆ ∈ Eˆ , yˆ ∈ F(xˆ). A pair (xˆ, yˆ) is called a Henig efficient
element of (VP) if there exists ε ∈ (, δ) such that
where δ := inf{ b : b ∈ B}, F(Eˆ ) =
Definition . (See []) The interior tangent cone IT(S, y¯) of S at y¯ is the set of all y ∈ Y
such that for any tn → + and yn → y, we have y¯ + tnyn ∈ S.
IT(S, y¯) = IT(int S, y¯) = intcone(S – y¯).
Theorem . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D),
(uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), F is second-order lower radial composed derivable at (xˆ, yˆ)
in the direction (uˆ , vˆ), and G is second-order radial composed derivable at (xˆ, zˆ) in the
direction (uˆ , wˆ ). Then there exists εˆ ∈ (, δ) such that
On the contrary, suppose that (.) does not hold. Then there exist x¯ ∈ dom Rl F(xˆ, yˆ,
uˆ , vˆ) ∩ dom R G(xˆ, zˆ, uˆ , wˆ ), y¯ ∈ Rl F(xˆ, yˆ, uˆ , vˆ)(x¯), and z¯ ∈ R G(xˆ, zˆ, uˆ, wˆ )(x¯) such that
z¯ ∈ – int D.
Since (u , w ) R(
n n ∈
k k k
G, (x, z)), there exist sequences t > and (x , z )
ˆ ˆ
n n n
G such that
Hence, there exist t > and (u , w ) R(
n n n ∈
G, (x, z)) such that
ˆ ˆ
t (u , w ) – (u, w)
n n n ˆ ˆ
t (w – w) –
n n ˆ ∈
D is a cone, we obtain
From (.) it follows that there exists N
∈
N such that
–D and –D is a convex cone, it follows that
n ∈
D – D = –
N such that
n > N , k > K (n).
∀
D is a cone, we obtain
n > N , k > K (n).
∀
–D and –D is a convex cone, it follows that
D – D = –
n > N , k > K (n).
∀
D – D = –
G, we obtain z
k k
G(x ) + D. Hence, there exists z
¯ ∈
n n
k k
Eˆ . It follows from (.) that t (x – x)
ˆ →
n n
→ ∞
, and hence, u
n ∈
R(Eˆ , x). It follows from (.) that t (u – u) x, and hence,
ˆ n n ˆ → ¯
R(R(Eˆ , x), u). By Proposition ., since y
ˆ ˆ ¯ ∈
R F(x, y, u, v)(x), we conclude that
ˆ ˆ ˆ ˆ ¯
l
R F(x, y, u, v) R R(Eˆ , x), u
ˆ ˆ ˆ ˆ ˆ ˆ
l
clcone clcone
F(Eˆ ) + C – y – v .
ˆ ˆ
From (.) it follows that
clcone clcone
F(Eˆ ) + C – y – v
ˆ ˆ ∩
intcone ε
( U + B) = .
∅
In the similar way, we conclude that
This is a contradiction to (.). The proof is completed.
Corollary . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D),
(uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), F is second-order lower radial composed derivable at (xˆ, yˆ) in
the direction (uˆ , vˆ), and G is second-order lower radial composed derivable at (xˆ, zˆ) in the
direction (uˆ , wˆ). Then there exists a number εˆ ∈ (, δ) such that
Proof The proof follows directly from Theorem . and Remark .(ii).
Example . Let R be the set of real numbers, X = Y = Z = R, C = D = {t : t ≥ }, B = {}.
Define the set-valued maps F : X → Y and G : X → Z by
F(x) = G(x) =
{y : y ≥ } if x ≥ ,
{y : y ≥ x} otherwise.
zˆ ∈ G() ∩ (–D) = {},
R epi G, (, ) = (x, y) : x ∈ R, y ≥ ,
Rl F(, , , )(x) = Rl G(, , , )(x) =
R G(, , , )(x) = [, +∞), x ∈ R,
Then, the inclusions of Theorem . and Corollaries . and . are true.
Theorem . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D),
(uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), F is second-order lower radial derivable at (xˆ, yˆ) in the
direction (uˆ , vˆ), and G is second-order radial derivable at (xˆ, zˆ) in the direction (uˆ , wˆ ). Then
there exists a number εˆ ∈ (, δ) such that
RlF(xˆ, yˆ, uˆ , vˆ)(x), RG(xˆ, zˆ, uˆ , wˆ )(x) ∩
Proof On the contrary, suppose that (.) does not hold. Then, for any ε ∈ (, δ), there exist
x¯ ∈ dom RlF(xˆ, yˆ, uˆ , vˆ) ∩ dom RG(xˆ, zˆ, uˆ , wˆ ), y¯ ∈ RlF(xˆ, yˆ, uˆ , vˆ)(x¯), and z¯ ∈ RG(xˆ, zˆ, uˆ , wˆ )(x¯)
such that
R G(x, z, u, w)(x) it follows that
ˆ ˆ ˆ ˆ ¯
Hence, there exist t > , x
n
x, and z
z such that
x + t u +
ˆ n ˆ
t x , z + t w + t z
n ˆ n ˆ
n n
n ∈
n ∈
The set of positive integers is denoted by N . From (.) and z
z it follows that there
∈
N such that
n ∩
x + t u +
ˆ n ˆ
n ∈
n ∈
D and –D are convex cones, we obtain
n ∈
–D – D –
D = –
It follows from (.) that there exists z
˜n ∈
n ∈ {˜n}
Since (.) and D is a convex cone, we obtain
D – D = –
x it follows that x
¯ ∈
R (Eˆ , x, u). By Proposition . and y
ˆ ˆ
¯ ∈
R F(x, y, u, v) R (Eˆ , x, u)
ˆ ˆ ˆ ˆ ˆ ˆ
l
clcone cone
F(Eˆ ) + C – y – v .
ˆ ˆ
It follows from (.) that
intcone ε
( U + B) is open, we obtain
clcone cone
F(Eˆ ) + C – y – v
ˆ ˆ ∩
cone cone
F(Eˆ ) + C – y – v
ˆ ˆ ∩
cone ε
( U + B) is a pointed cone, it follows that
F(Eˆ ) + C – y – v
ˆ ˆ ∩
F(Eˆ ) + C – y v –
ˆ ∩ ˆ
In a similar way, we conclude that
cone ε
( U + B) is a pointed cone, we obtain
ˆ ∈
Corollary . Suppose that (x, y) is a Henig efficient element of (VP), z
ˆ ˆ ˆ ∈
(–C) (–D), F is second-order lower radial derivable at (x, y) in the direction
× ˆ ˆ
there exists a number
R F(x, y, u, v)(x), R G(x, z, u, w)(x)
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
l l
D) =
for all x
Proof The proof follows immediately from Theorem . and Remark .(ii).
Corollary . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D),
(uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), B is a base of C, F is second-order lower radial derivable at
(xˆ, yˆ) in the direction (uˆ , vˆ), and G is second-order lower radial derivable at (xˆ, zˆ) in the
direction (uˆ , wˆ). Then there exists a number εˆ ∈ (, δ) such that
Proof It is similar to the proof of Corollary ..
Example . Let R be the set of real numbers, X = Y = Z = R, C = D = {t : t ≥ }, and
B = {}. Define the set-valued maps F : X → Y and G : X → Z by
F(x) = {y : y ≥ }, x ∈ R,
G(x) = {y : y ≥ x}, x ∈ R.
zˆ ∈ G() ∩ (–D) = {},
Then, the inclusions of Theorem . and Corollaries . and . are true.
min f (x),
s.t. g(x) ∈ –D, x ∈ X.
Similarly to Definition . in [], we introduce the following second-order generalized
lower (upper) directional derivative for vector-valued functions.
Remark . When the set-valued map F becomes to a vector-valued function f , which is
Fréchet differentiable at xˆ, letting vˆ := f (xˆ)uˆ , we have
Remark . When the set-valued map F becomes to a vector-valued function f , which is
Fréchet differentiable at xˆ, letting vˆ := f (xˆ)uˆ , we have
Corollary . Suppose that (xˆ, yˆ) is a Henig efficient element of (P) and g(xˆ) ∈ –D. Then
there exists a number εˆ ∈ (, δ) such that
D˜lf+(xˆ, uˆ )(x), D˜g+(xˆ, uˆ )(x) ∩
5 Conclusions
In this paper, we introduced some new kinds of lower radial tangent cone, second-order
lower radial tangent set, and second-order radial tangent set. By virtue of these concepts,
second-order radial composed tangent derivative, second-order radial tangent
derivative, second-order lower radial composed tangent derivative, and second-order lower
radial tangent derivative for a set-valued map are introduced. Compared with the
secondorder composed contingent derivative D F(xˆ, yˆ, uˆ , vˆ) introduced in [, ], the
secondorder contingent derivative DF(xˆ, yˆ, uˆ , vˆ), second-order radial composed tangent
derivative R F(xˆ, yˆ, uˆ , vˆ), and second-order radial tangent derivative RF(xˆ, yˆ, uˆ , vˆ), second-order
lower radial composed tangent derivative Rl F(xˆ, yˆ, uˆ , vˆ), and second-order lower radial
tangent derivative RlF(xˆ, yˆ, uˆ , vˆ) have nice properties:
Rl F(xˆ, yˆ, uˆ , vˆ) R R(E, xˆ), uˆ
⊂ clcone clcone F(E) + C – yˆ – vˆ
derivatives of the objective function and constraint function are separated.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
Authors’ information
Yihong Xu (1969-), Professor, Doctor, the major field of interest is the set-valued optimization.
Acknowledgements
This research was supported by the National Natural Science Foundation of China Grant 11461044 and the Natural
Science Foundation of Jiangxi Province (20151BAB201027).
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